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Enforce the norm of the addition of two vectors plus scalar

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I have two vectors of the same dimension: and and a scalar β. I'm looking for the value of β such that .
Do you have an idea how to find β ?


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Accepted Answer

Mohammad Alhashash
Mohammad Alhashash on 5 Jun 2019
I assumed that a and b are 3 elements-vector and then solved the equation beta analytically.
a = [1 2 3];
b = [2 3 4];
beta = roots([b(1)^2+b(2)^2+b(3)^2 , 2*(a(1)*b(1)+a(2)*b(2)+a(3)*b(3)),a(1)^2+a(2)^2+a(3)^2-1])
the result is two solution corresponding to two different value of beta.
to verify our solution, simply try it;
or :

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Thomas Barbier
Thomas Barbier on 5 Jun 2019
This solution exists though if , otherwise no (real) solution is available.

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